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Convergence of the Weil-Petersson metric on the Teichmüller space of bordered Riemann surfaces
Aalto Univ, Dept Math & Syst Anal, POB 11100, FI-00076 Aalto, Finland.
Univ Manitoba, Dept Math, Winnipeg, MB R3T 2N2, Canada.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2017 (English)In: Communications in Contemporary Mathematics, ISSN 0219-1997, Vol. 19, no 1, 1650025Article in journal (Refereed) Published
Abstract [en]

Consider a Riemann surface of genus g bordered by n curves homeomorphic to the unit circle, and assume that 2g−2+n > 0. For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space.We show that any tangent vector can be represented as the derivative of a holomor-phic curve whose representative Beltrami differentials are simultaneously in L2 and L ,and furthermore that the space of (−1,1) differentials in L2 ∩L decomposes as a directsum of infinitesimally trivial differentials and L2 harmonic (−1,1) differentials. Thus thetangent space of this Teichmüller space is given by L2 harmonic Beltrami differentials.We conclude that this Teichmüller space has a finite Weil-Petersson Hermitian metric.Finally, we show that the aforementioned Teichmüller space is locally modeled on a spaceof L2 harmonic Beltrami differentials.

Place, publisher, year, edition, pages
World Scientific, 2017. Vol. 19, no 1, 1650025
Keyword [en]
Weil-Petersson metric, L-2 Beltrami differentials, bordered Riemann surfaces, Teichmuller theory, infinitesimally trivial Beltrami differentials, Gardiner-Schiffer variation
National Category
Geometry
Identifiers
URN: urn:nbn:se:uu:diva-284916DOI: 10.1142/S0219199716500255ISI: 000389231700008OAI: oai:DiVA.org:uu-284916DiVA: diva2:920966
Available from: 2016-04-19 Created: 2016-04-19 Last updated: 2017-11-30Bibliographically approved

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Staubach, Wolfgang

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